Problem: What is the units digit of $31^3+13^3$?
The units digit of $31^3$ is the same as the units digit of $1^3$, which is 1.  The units digit of $13^3$ is the same as the units digit of $3^3$, which is 7.  Therefore, the units digit of $31^3+13^3$ is $\boxed{8}$.